Shattered Iterations

TL;DR

This paper introduces shattered iterations, a novel form of iterated forcing adding random reals, and applies them to establish new consistency results related to cardinal invariants in set theory.

Contribution

It develops the theory of shattered iterations, including amalgamated limits of cBa's and preservation theorems, and applies these to solve longstanding questions in cardinal invariants.

Findings

Established the consistency of  < cov(meager) < non(meager)

Introduced the concept of shattered iterations and amalgamated limits of cBa's

Proved preservation theorems for finitely additive measures on cBa's

Abstract

We develop iterated forcing constructions dual to finite support iterations in the sense that they add random reals instead of Cohen reals in limit steps. In view of useful applications we focus in particular on two-dimensional "random" iterations, which we call shattered iterations. As basic tools for such iterations we investigate several concepts that are interesting in their own right. Namely, we discuss correct diagrams, we introduce the amalgamated limit of cBa's, a construction generalizing both the direct limit and the two-step amalgamation of cBa's, we present a detailed account of cBa's carrying finitely additive strictly positive measures, and we prove a general preservation theorem for such cBa's in amalgamated limits. As application, we obtain new consistency results on cardinal invariants in Cichon's diagram. For example, we show the consistency of aleph_1 < cov (meager) < non (meager), thus answering an old question of A. Miller.